Can you work out the area of the inner square and give an
explanation of how you did it?

Hidden Squares

Stage: 3 Challenge Level:

A neat solution to the first part of the problem was received
from Sana, Jenny, Chris and Rosion of Madras College, St. Andrews.
I like the use of the lines rather than thinking about the
rectangles in the same way as the squares (which is what I did).
This idea can be generalised quite easily to an n x n chess
board.

Other correct solutions were received from Andrei of School 205
Bucharest, Mary of Birchwood Community High School and Chen of The
Chinese High School, Singapore.

Well done to all of you.

There are 1296 different rectangles on the chess board.

204 of these rectangles are squares.

First looking at the squares:

Consider placing a square of size 1 x 1 along the left hand edge
of the chessboard. This square can be in any one of 8 positions (as
there are 8 by 8 squares on a chessboard). Similarly, the square
can be placed in any one of eight positions along the top edge. So
the total number of

1 x 1 squares = 8 x 8 = 64.

A 2 x 2 square can occupy a 7 positions along the left hand edge
and 7 positions along the top edge 7, giving 7 x 7 = 49 squares of
size 2 x 2.

Continuing in this way we get squares of size 3 x 3, 4 x 4, and
so on.

Size of Square

Number of squares

1 x 1

8 x 8 = 64

2 x 2

7 x 7 = 49

3 x 3

6 x 6 = 36

4 x 4

5 x 5 = 25

5 x5

4 x 4 = 16

6 x 6

3 x 3 = 9

7 x 7

2 x 2 = 4

8 x 8

1 x 1 = 1

So there are 204 squares.

Then looking at rectangles:

There are 9 vertical lines and 9 horizontal lines on the chess
board.

To form a rectangle you must choose 2 of the 9 vertical lines
and 2 of the 9 horizontal lines.

For the two horizontal lines: the first line can be chosen in 9
ways the second in eight ways. This would imply that you could tell
the difference between lines 1 and 3 say and 3 and 1, which is not
the case so you need to divide 9 x8 8 by 2, making 36.

Similarly you can choose the two vertical lines in 36 ways.

So the number of rectangles is given by 362

362 = 1296

The following approach utilises several of the other solutions
sent in.

A rectangle (or square) will have a height between 1 and 8 units
and a width between 1 and 8 units. Tis can be represented by a
table with each possible width represented by a column and each
eight by a row.

The entries in the table below then indicate the number of each
size rectangle on the chessboard (using similar arguments to first
part of the problem above).

The final column of the table gives the total number of
rectangles in each row.

Height/Width

1

2

3

4

5

6

7

8

Total

1

8 x 8

7 x 8

6 x 8

5 x 8

4 x 8

3 x 8

2 x 8

1 x 8

8 x(1+2+3+4+5+6+7+8}= 8 x 36

2

8 x 7

7 x 7

6 x 7

5 x 7

4 x 7

5 x 7

2 x 7

1 x 7

7 x(1+2+3+4+5+6+7+8}= 7 x 36

3

8 x 6

7 x 6

6 x 6

5 x 6

4 x 6

3 x 6

2 x 6

1 x 6

6 x(1+2+3+4+5+6+7+8}= 6 x 36

4

8 x 5

7 x 5

6 x 5

5 x 5

4 x 5

3 x 5

2 x 5

1 x 5

5 x(1+2+3+4+5+6+7+8}

5

8 x 4

7 x 4

6 x 4

4 x 4

5 x 4

3 x 4

2 x 4

1 x 4

4 x(1+2+3+4+5+6+7+8}

6

8 x 3

7 x 3

6 x 3

4 x 3

5 x 3

3 x 3

2 x 3

1 x 3

3 x(1+2+3+4+5+6+7+8}

7

8 x 2

7 x 2

6 x 2

4 x 2

5 x 2

3 x 2

2 x 2

1 x 2

2 x(1+2+3+4+5+6+7+8}

8

8 x 1

7 x 1

6 x 1

4 x 1

5 x 1

3 x 1

2 x 1

1 x 1

1 x(1+2+3+4+5+6+7+8}

(1+2+3+4+5+6+7+8}x 36 = 362

Therefore the total number of rectangles in an 8 x 8 grid is (1
+ 2 + 3 + ... + 8)2

The total number of rectangles in an n x n grid is (1 + 2 + 3 +
... + n)2 = (n2 (n+1)2 )/4. This uses
the formula for the sum of the first n natural numbers, an
arithmetic progression. For more details of this look at the proof
sorter (link).

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